 19.19.1: In each of 1 through 10, carry out the indicated calculation.(3 4i)...
 19.19.35: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.48: In each of 1 through 10, write the function value in the form a + b...
 19.19.66: In each of 1 through 6, determine all values of the complex logarit...
 19.19.74: In each of 1 through 14, determine all values of zw.i 1+i
 19.19.2: In each of 1 through 10, carry out the indicated calculation.i(6 2i...
 19.19.36: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.49: In each of 1 through 10, write the function value in the form a + b...
 19.19.67: In each of 1 through 6, determine all values of the complex logarit...
 19.19.75: In each of 1 through 14, determine all values of zw.(1 + i)2
 19.19.3: In each of 1 through 10, carry out the indicated calculation.(2 + i...
 19.19.37: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.50: In each of 1 through 10, write the function value in the form a + b...
 19.19.68: In each of 1 through 6, determine all values of the complex logarit...
 19.19.76: In each of 1 through 14, determine all values of zw.ii
 19.19.4: In each of 1 through 10, carry out the indicated calculation.((2 + ...
 19.19.38: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.51: In each of 1 through 10, write the function value in the form a + b...
 19.19.69: In each of 1 through 6, determine all values of the complex logarit...
 19.19.77: In each of 1 through 14, determine all values of zw.. (1 + i)2i
 19.19.5: In each of 1 through 10, carry out the indicated calculation.(17 6i...
 19.19.39: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.52: In each of 1 through 10, write the function value in the form a + b...
 19.19.70: In each of 1 through 6, determine all values of the complex logarit...
 19.19.78: In each of 1 through 14, determine all values of zw.(1 + i)3i
 19.19.6: In each of 1 through 10, carry out the indicated calculation.3i/(4...
 19.19.40: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.53: In each of 1 through 10, write the function value in the form a + b...
 19.19.71: In each of 1 through 6, determine all values of the complex logarit...
 19.19.79: In each of 1 through 14, determine all values of zw.(1 i)1/3
 19.19.7: In each of 1 through 10, carry out the indicated calculation.i 3 4i...
 19.19.41: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.54: In each of 1 through 10, write the function value in the form a + b...
 19.19.72: Let z and w be nonzero complex numbers. Show that each value of log...
 19.19.80: In each of 1 through 14, determine all values of zw.i 1/4
 19.19.8: In each of 1 through 10, carry out the indicated calculation.(3 + i)3
 19.19.42: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.55: In each of 1 through 10, write the function value in the form a + b...
 19.19.73: Let z and w be nonzero complex numbers. Show that each value of log...
 19.19.81: In each of 1 through 14, determine all values of zw.161/4
 19.19.9: In each of 1 through 10, carry out the indicated calculation.((6 + ...
 19.19.43: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.56: In each of 1 through 10, write the function value in the form a + b...
 19.19.82: In each of 1 through 14, determine all values of zw.(4)2i
 19.19.10: In each of 1 through 10, carry out the indicated calculation.(3 8i)...
 19.19.44: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.57: In each of 1 through 10, write the function value in the form a + b...
 19.19.83: In each of 1 through 14, determine all values of zw.623i
 19.19.11: In each of 11 through 16, determine the magnitudeand all of the arg...
 19.19.45: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.58: Determine u and v such that ez2 = u(x, y) + iv(x, y). Show that u a...
 19.19.84: In each of 1 through 14, determine all values of zw.(16)1/4
 19.19.12: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.46: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.59: Find u and v such that e1/z =u(x, y)+iv(x, y). Show that u and v sa...
 19.19.85: In each of 1 through 14, determine all values of zw.][(1 + i)/(1 i)...
 19.19.13: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.47: Let zn = an + ibn be a sequence of complex numbers. We say that thi...
 19.19.60: Find u and v such that zez = u(x, y) +iv(x, y). Show that u and v s...
 19.19.86: In each of 1 through 14, determine all values of zw.11/6/2017
 19.19.14: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.61: Find u and v such that cos2 (z) = u(x, y) + iv(x, y). Show that u a...
 19.19.87: In each of 1 through 14, determine all values of zw.(7i)3i
 19.19.15: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.62: Find all solutions of ez = 2i.
 19.19.88: Let u1, , un be the nth roots of unity with n a positive integer an...
 19.19.16: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.63: Derive the following identities. (a) sin(z + w) = sin(z) cos(w) + c...
 19.19.89: Let n be a positive integer, and let =e2i/n . Evaluate n1 j=0(1)j j .
 19.19.17: In each of 17 through 22, write the number in polar form2 + 2i
 19.19.64: Find all solutions of ez = 2.
 19.19.18: In each of 17 through 22, write the number in polar form7i
 19.19.65: Find all solutions of sin(z) = i.
 19.19.19: In each of 17 through 22, write the number in polar form5 2i
 19.19.20: In each of 17 through 22, write the number in polar form4 i
 19.19.21: In each of 17 through 22, write the number in polar form8 + i
 19.19.22: In each of 17 through 22, write the number in polar form12 + 3i
 19.19.23: Show that, for any positive integer n, i 4n = 1,i 4n+1 = i,i 4n+2 =...
 19.19.24: Let z = a + ib. Determine Re(z2 ) and Im(z2 )
 19.19.25: Show that complex numbers z, w, and u form vertices of an equilater...
 19.19.26: Show that z2 = (z)2 if and only if z is either real or pure imaginary.
 19.19.27: Let z and w be numbers with zw = 1. Suppose either z or w has magni...
 19.19.28: Show that, for any numbers z and w, z + w 2 + z w 2 = 2 z 2 +...
 19.19.29: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.30: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.31: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.32: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.33: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.34: In each of 29 through 37, a set of complex numbers is specified. De...
Solutions for Chapter 19: Complex Numbers and Functions
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 19: Complex Numbers and Functions
Get Full SolutionsChapter 19: Complex Numbers and Functions includes 89 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Since 89 problems in chapter 19: Complex Numbers and Functions have been answered, more than 7773 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.